I see no mechanic stating that fighters don't have that special ability. If he wants to throw the sword 100 yards at an orc, he's in trouble due to the thrown mechanics. If he wants to use the Moon Split special ability that the rules don't say that he doesn't have, that sword flies all the way to the moon and cracks it in half without dealing damage. That's how this ability that the rules don't say he doesn't have works.
No, that's not how a rule system works. The rule system details what entity X can do not exhaustively present everything it cannot.
That's not entirely accurate. Much of the lore would remain the same, but a lot would change. For example, if Pendragon doesn't have all the Mordenkainen, Bigby, Rary, etc. spells, then those suddenly vanish from the lore. Mechanics can also create lore and vice versa.
Mordenkainen, Bigby, et al. are characters that can be represented by (almost) any rules system. Would the expressions in any system be able to do everything and only what (any particular) D&D edition expression be able to do? Probably not. But, the characters would remain.
1e I think, but not sure.
Edit: Yes. 1e. I looked it up and it's in the DMG. Magic is powered by the positive and negative planes.
Well, it sort of says that if you squint (which wasn't your original assertion anyway). It says spells are charged with energy for one or both those planes which allows the casting but the actual spell effect is powered from energy from "some plane in the multiverse". That has little to no bearing on whether other external factors (like moon phase) affect spell casting. Perhaps, the planar boundaries change permeability as the moons wax and wane so the amount of energy available changes resulting in changes to effective caster levels and saving throw chances. It isn't against lore; lore is silent on the subject which is my original point.